cp's OEIS Frontend

A358247 Number of n-regular, N_0-weighted pseudographs on 2 vertices with total edge weight 7, up to isomorphism.

%I A358247 #21 Dec 01 2022 10:22:51
%S A358247 1,8,28,71,132,217,309,417,521,638,746,866,975,1096,1205,1326,1435,
%T A358247 1556,1665,1786,1895,2016,2125,2246,2355,2476,2585,2706,2815,2936,
%U A358247 3045,3166,3275,3396,3505,3626,3735,3856,3965,4086,4195,4316,4425,4546,4655
%N A358247 Number of n-regular, N_0-weighted pseudographs on 2 vertices with total edge weight 7, up to isomorphism.
%C A358247 Pseudographs are finite graphs with undirected edges without identity, where parallel edges between the same vertices and loops are allowed.
%H A358247 Lars Göttgens, <a href="/A358247/b358247.txt">Table of n, a(n) for n = 1..10000</a>
%H A358247 J. Flake and V. Mackscheidt, <a href="https://arxiv.org/abs/2206.08226">Interpolating PBW Deformations for the Orthosymplectic Groups</a>, arXiv:2206.08226 [math.RT], 2022.
%H A358247 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Pseudograph.html">Pseudograph</a>.
%e A358247 For n = 2 the a(2) = 8 such pseudographs are: 1. two vertices connected by a 7-edge and a 0-edge, 2. two vertices connected by a 6-edge and a 1-edge, 3. two vertices connected by a 5-edge and a 2-edge, 4. two vertices connected by a 4-edge and a 3-edge, 5. two vertices where one has a 7-loop and the other one has a 0-loop, 6. two vertices where one has a 6-loop and the other one has a 1-loop, 7. two vertices where one has a 5-loop and the other one has a 2-loop, 8. two vertices where one has a 4-loop and the other one has a 3-loop.
%o A358247 (Julia)
%o A358247 using Combinatorics
%o A358247 function A(n::Int)
%o A358247     sum_total = 7
%o A358247     result = 0
%o A358247     for num_loops in 0:div(n, 2)
%o A358247         num_cross = n - 2 * num_loops
%o A358247         for sum_cross in 0:sum_total
%o A358247             for sum_loop1 in 0:sum_total-sum_cross
%o A358247                 sum_loop2 = sum_total - sum_cross - sum_loop1
%o A358247                 if sum_loop2 == sum_loop1
%o A358247                     result +=
%o A358247                         div(
%o A358247                             npartitions_with_zero(sum_loop2, num_loops) *
%o A358247                             (npartitions_with_zero(sum_loop2, num_loops) + 1),
%o A358247                             2,
%o A358247                         ) * npartitions_with_zero(sum_cross, num_cross)
%o A358247                 elseif sum_loop2 > sum_loop1
%o A358247                     result +=
%o A358247                         npartitions_with_zero(sum_loop2, num_loops) *
%o A358247                         npartitions_with_zero(sum_loop1, num_loops) *
%o A358247                         npartitions_with_zero(sum_cross, num_cross)
%o A358247                 end
%o A358247             end
%o A358247         end
%o A358247     end
%o A358247     return result
%o A358247 end
%o A358247 function npartitions_with_zero(n::Int, m::Int)
%o A358247     if m == 0
%o A358247         if n == 0
%o A358247             return 1
%o A358247         else
%o A358247             return 0
%o A358247         end
%o A358247     else
%o A358247         return Combinatorics.npartitions(n + m, m)
%o A358247     end
%o A358247 end
%o A358247 print([A(n) for n in 1:45])
%Y A358247 Other total edge weights: 3 (A358243), 4 (A358244), 5 (A358245), 6 (A358246), 8 (A358248), 9 (A358249).
%K A358247 nonn
%O A358247 1,2
%A A358247 _Lars Göttgens_, Nov 04 2022